A discussion of the number of degrees of freedom, and their dynamical
properties, in higher-derivative gravitational theories is presented. The
complete non-linear sigma model for these degrees of freedom is exhibited using
the method of auxiliary fields. As a by-product we present a consistent
non-linear coupling of a spin-2 tensor to gravitation. It is shown that
non-vanishing $(C_{\mu\nu\alpha\beta})^{2}$ terms arise in $N=1$, $D=4$
superstring Lagrangians due to one-loop radiative corrections with light field
internal lines.Comment: 8 pages, LaTeX2e and AMS-LaTeX 1.2, Talk presented at SUSY'95, Ecole
  Polytechnique in Palaiseau, Palaiseau, France, May 15-19, 199

Hindawi, Ahmed

Ovrut, Burt A.

Waldram, Daniel

English

arXiv

A discussion of the number of degrees of freedom, and their dynamical properties, in higher-derivative gravitational theories is presented. The complete non-linear sigma model for these degrees of freedom is exhibited using the method of auxiliary fields. As a by-product we present a consistent non-linear coupling of a spin-2 tensor to gravitation. It is shown that non-vanishing (C_{\mu\nu\alpha\beta})^2 terms arise in N=1, D=4 superstring Lagrangians due to one-loop radiative corrections with light field internal lines

Hindawi, A A

Ovrut, B A

Waldram, D

CERN Document Server

H
EP
-T
H
-9
50
61
73
UPR-672-T
Higher-Derivative Gravitation in Bosonic
and Superstring Theories
Ahmed Hindawi, Burt A. Ovrut, and Daniel Waldram

Department of Physics, University of Pennsylvania
Philadelphia, PA 19104-6396, USA
Abstract
A discussion of the number of degrees of freedom, and their dynamical properties,
in higher-derivative gravitational theories is presented. The complete non-linear sigma
model for these degrees of freedom is exhibited using the method of auxiliary elds.
As a by-product we present a consistent non-linear coupling of a spin-2 tensor to
gravitation. It is shown that non-vanishing (C

)
2
terms arise in N = 1, D = 4
superstring Lagrangians due to one-loop radiative corrections with light eld internal
lines.
1 Bosonic Gravitation
The usual Einstein theory of gravitation involves a symmetric tensor g

whose dynamics is
determined by the Lagrangian
L =
1
2
2
R (1)
The dieomorphic gauge group reduces the number of degrees of freedom from ten down to
six. Einstein's equations further reduce the degrees of freedom to two, which correspond to
a physical spin-2 massless graviton. Now let us consider an extension of Einstein's theory by
including terms in the action which are quadratic in the curvature tensors. This extended
Lagrangian is given by
L =
1
2
2
R+ R
2
+ (C

)
2
+ (R

)
2
(2)
where R
2
, (C

)
2
, and (R

)
2
are a complete set of CP-even quadratic curvature terms.
The topological Gauss-Bonnet term is given by
GB = (C

)
2
  2(R

)
2
+
2
3
R
2
(3)

Talk presented at SUSY'95, Ecole Polytechnique in Palaiseau, Palaiseau, France, May 15-19, 1995. Work
supported in part by DOE Contract DOE-AC02-76-ERO-3071 and NATO Grant CRG. 940784.
1
Therefore, we can write
L =
1
2
2
R + aR
2
  b(C

)
2
+ cGB (4)
In this case, it can be shown [1] that there is still a physical spin-2 massless graviton in the
spectrum. However, the addition of the R
2
term introduces a new physical spin-0 scalar,
, with mass m = 1=
p
12a
2
. Similarly, the (C

)
2
term introduces a spin-2 symmetric
tensor, 

, with massm = 1=
p
4b
2
but this eld, having wrong sign kinetic energy, is ghost-
like. The GB term, being a total divergence, is purely topological and does not lead to any
new degrees of freedom. The scalar  is perfectly physical and can lead to very interesting
new physics [2]. The new tensor 

, however, appears to be problematical. There have
been a number of attempts to show that the ghost-like behavior of 

is illusory, being
an artifact of linearization [3]. Other authors have pointed out that since the mass of 

is near the Planck scale, other Planck scale physics may come in to correct the situation
[4]. In all these attempts, the gravitational theories being discussed were not necessarily
consistent and well dened. However, in recent years, superstring theories have emerged
as nite, unitary theories of gravitation. Superstrings, therefore, are an ideal laboratory
for exploring the issue of the ghost- like behavior of 

, as well as for asking whether the
scalar  occurs in the superstring Lagrangian. Hence, we want to explore the question \Do
quadratic gravitation terms appear in the N = 1, D = 4 superstring Lagrangian?"
Before doing this, however, we would like to present further details about the emergence
of the new degrees of freedom in quadratic gravitation. We begin by adding to Einstein
gravitation, quadratic terms associated with the scalar curvature only. That is, we consider
the action
S =
Z
d
4
x
p
 g

R+
1
6
m
 2
R
2

(5)
The equations of motion derived from this action are of fourth order and their physical
meaning is somewhat obscure. These equations can be reduced to second order, and their
physical content illuminated, by introducing an auxiliary eld . The action then becomes
S =
Z
d
4
x
p
 g

R +
1
6
m
 2
R
2
 
1
6
m
 2
h
R  3m
2
n
e

  1
oi
2

(6)
=
Z
d
4
x
p
 g

e

R 
3
2
m
2
h
e

  1
i
2

(7)
Note that the  equation of motion sets the square bracket in equation (6) to zero. Hence,
action (7) with the auxiliary eld  is equivalent to the original action (5). Now, let us
perform a Weyl rescaling of the metric
g

= e
 
g

(8)
It follows that
p
 g = e
 2
p
 g (9)
2
R = e


R+ 3r
2
 
3
2
h
r
i
2

(10)
where r

g

= 0. Therefore,
p
 ge

R =
p
 g

R+ 3r
2
 
3
2
h
r
i
2

 
3
2
m
2
p
 g

e

  1

2
=  
3
2
m
2
p
 g

1   e
 

2
(11)
and the action becomes
S =
Z
d
4
x
p
 g

R 
3
2
h
r
i
2
 
3
2
m
2
h
1   e
 
i
2

(12)
where we have dropped a total divergence term. It follows that the higher-derivative pure
gravity theory described by action (5) is equivalent to a theory of normal Einstein gravity
coupled to a real scalar eld . It is important to note that, with respect to the metric
signature ( ;+;+;+) we are using, the kinetic energy term for  has the correct sign and,
hence, that  is not ghost like. Also, note that a unique potential energy function
V () =
3
2
m
2

1   e
 

2
(13)
emerges which has a stable minimum at  = 0. We conclude that R+R
2
gravitation with
metric g

is equivalent to R gravitation with metric g

plus a non-ghost real scalar eld 
with a xed potential energy and a stable vacuum state. The property that  is non-ghost
like is suciently important that we will present yet another proof of this fact. This proof
was rst presented in [2]. If we expand the metric tensor as
g

= 

+ h

(14)
then the part of action (5) quadratic in h

is given by
S =
Z
d
4
x

1
4
h


r
2
n
P
(2)

  2P
(0)

o
+ 2m
 2

r
2

2
P
(0)


h


(15)
where P
(2)

and P
(0)

are transverse projection operators for h

. Inverting the kernel yields
the propagator

 1

=

r
2
P
(2)

+ 2m
 2
r
2
h
r
2
 m
2
i
P


 1
=
1
r
2

P
(2)

 
1
2
P
(0)


+
1
2(r
2
 m
2
)
P
(0)

(16)
The term proportional to (r
2
)
 1
corresponds to the usual two helicity massless graviton.
However, the term proportional to (r
2
 m
2
)
 1
represents the propagation of a real scalar
eld with positive energy and, hence, not a ghost. This corresponds to the results obtained
using the auxiliary eld above. We would like to point out that there may be very interesting
3
physics associated with the scalar eld . For example, as emphasized in [5],  may act as
a natural inaton in cosmology of the early universe.
Now let us consider Einstein gravity modied by quadratic terms involving the Weyl
tensor only. That is, consider the action
S =
Z
d
4
x
p
 g

R 
1
2
m
 2
C

C


(17)
Using the identity
C

C

= GB + 2

R

R

 
1
3
R
2

(18)
where GB is the topological Gauss-Bonnet combination dened in (3), the action becomes
S =
Z
d
4
x
p
 g

R m
 2

R

R

 
1
3
R
2

(19)
where we have dropped a total divergence. The fourth order equations of motion can be
reduced to second order equations by introducing an auxiliary symmetric tensor eld 

.
Using this eld, the action can be written as
S =
Z
d
4
x
p
 g
 
R G



+
m
2
4
h




  
2
i
!
(20)
where  = 

g

and G

= R

 
1
2
g

R is the Einstein tensor. Note that the 

equation
of motion is


= 2m
 2

R

 
1
6
g

R

(21)
Substituting this into (20) gives back the original action (19). As it stands, action (20) is
somewhat obscure since the G



term mixes g

and 

at the quadratic level. They
can, however, be decoupled by a eld redenition. First write the above action as
S =
Z
d
4
x
p
 g
 

1 +
1
2


g

  


R

+
m
2
4
h




  
2
i
!
(22)
Now transform the metric as
p
 gg

=
p
 g

1 +
1
2


g

  


g


(23)
or, equivalently
g

= (detA)
 1=2
A


g

A


=

1 +
1
2





  


(24)
Under this transformation
R

= R

 r

C


+r

C


+ C


C


  C


C


(25)
4
where r

g

= 0 and
C


=
1
2

X
 1



r

X

+r

X

 r

X


X

= g

= (detA)
 1=2
A


g

(26)
Inserting these transformations into the above and dropping a total divergence, the action
becomes [6]
S =
Z
d
4
x
p
 g
"
R + g


C


C


  C


C



 
m
2
4
(detA)
 1=2





  
2

#
(27)
Note that the action for 

is a complicated non-linear sigma model since C = C(X)
and X = X(). It is useful to consider the kinetic energy part of the action expanded to
quadratic order in 

only. It is found to be
S
quad

=
Z
d
4
x
p
 g

1
4
r



r



 
1
2
r



r



+
1
2
r



r

 
1
4
r

r



(28)
This action is clearly the curved space generalization of the Pauli-Fierz action for a spin-2
eld except that every term has the wrong sign! This implies, of course, that 

propa-
gates as a ghost. It is interesting to note that the action (27) is invariant under the gauge
transformation

0

= 

+r
(

)
  C




g
0

=

1 +
1
2

0

 1
"
det
1=2
A
0
det
1=2
A

1 +
1
2


g

+


0

  


#
(29)
This insures that the above action describes a consistent coupling of a spin-2 symmetric
tensor eld 

to Einstein gravitation at the full non-linear level [7]. We conclude, therefore,
that R + C
2
gravitation with metric g

is equivalent to R gravity with metric g

plus a
ghost-like symmetric tensor eld 

with a consistent non-linear coupling to gravity and
a xed potential energy. The physics in the eld 

is obscured by its ghost-like nature.
However, its ghost nature can be altered by yet higher-derivative terms, such as those one
would expect to nd generated in superstring theories. Therefore, at long last, we turn to
our discussion of quadratic supergravitation in superstring theory.
2 Superspace Formalism
In the Kahler (Einstein frame) superspace formalism, the most general Lagrangian for Ein-
stein gravity, matter and gauge elds is
L
E
=  
3
2
2
Z
d
4
E[K] +
1
8
Z
d
4

E
R
f(
i
)
ab
W
a
W

b
+ h.c. (30)
5
where we have ignored the superpotential term which is irrelevant for this discussion. The
fundamental supergravity superelds are R and W

, which are chiral, and G
 _
, which
is Hermitian. The bosonic R
2
, (C
mnpq
)
2
and (R
mn
)
2
terms are contained in the highest
components of the superelds

RR, (W

)
2
and (G
 _
)
2
respectively. One can also dene the
superGauss-Bonnet combination
SGB = 8(W

)
2
+ (

D
2
  8R)(G
2
 _
  4

RR) (31)
The bosonic Gauss-Bonnet term is contained in the highest chiral component of SGB. It
follows that the most general quadratic supergravity Lagrangian is given by
L
Q
=
Z
d
4
E

(


i
;
i
)

RR +
1
R
g(
i
)(W

)
2
+(


i
;
i
)(G
 _
)
2
+ h.c.

(32)
3 (2,2) Symmetric Z
N
Orbifolds
Although our discussion is perfectly general, we will limit ourselves to orbifolds, such as
Z
4
, which have (1; 1) moduli only. The relevant superelds are the dilaton, S, the diagonal
moduli T
II
, which we'll denote as T
I
, and all other moduli and matter superelds, which we
denote collectively as 
i
. The associated Kahler potential is
K = K
0
+ Z
ij


i

j
+O((

)
2
)

2
K
0
=   ln(S +

S) 
X
(T
I
+

T
I
)
Z
ij
= 
ij
Y
(T
I
+

T
I
)
q
i
I
(33)
The tree level coupling functions f
ab
and g can be computed uniquely from amplitude com-
putations and are given by
f
ab
= 
ab
k
a
S; g = S (34)
There is some ambiguity in the values of  and  due to the ambiguity in the denition of
the linear supermultiplet. We will take the conventional choice
 =  S;  = 4S (35)
It follows that, at tree level, the complete Z
N
orbifold Lagrangian is given by L = L
E
+ L
Q
where
L
Q
=
1
4
Z
d
4

E
R
S SGB + h.c. (36)
Using this Lagrangian, we now compute the one-loop moduli-gravity-gravity anomalous
threshold correction [8]. This must actually be carried out in the conventional (string frame)
6
superspace formalism and then transformed to Kahler superspace [9]. We also compute the
relevant superGreen-Schwarz graphs. Here we will simply present the result. We nd that
L
1-loop
massless
=
1
24(4)
2
X

h
I
Z
d
4
(

D
2
  8R)

RR
1
@
2
D
2
ln(T
I
+

T
I
)
+ (b
I
  8p
I
)
Z
d
4
(W

)
2
1
@
2
D
2
ln(T
I
+

T
I
) (37)
+p
I
Z
d
4
(8(W

)
2
+ (

D
2
  8R)((G
 _
)
2
  4

RR))
1
@
2
D
2
ln(T
I
+

T
I
) + h.c.

where
h
I
=
1
12
(3T + 3#
T
q
I
+ ')
b
I
= 21 + 1 + n
I
M
  dimG+
X
(1 + 2q
i
I
)  24
I
GS
p
I
=  
3
8
dimG  
1
8
 
1
24
X
1 +    3
I
GS
(38)
The coecients 
T
and #
T
, which arise from moduli loops, and ' and , which arise from
gravity and dilaton loops, are unknown. However, as we shall see, it is not necessary to know
their values to accomplish our goal. Now note that if h
I
6= 0 then there are non-vanishing
R
2
terms in the superstring Lagrangian. If b
I
  8p
I
6= 0 then the Lagrangian has C
2
terms.
Coecient p
I
6= 0 merely produces a Gauss-Bonnet term. With four unknown parameters
what can we learn? The answer is, a great deal! Let us take the specic example of the Z
4
orbifold. In this case, the Green-Schwarz coecients are known [10]

1;2
GS
=  30; 
3
GS
= 0 (39)
which gives the result
b
1;2
= 0; b
3
= 11  24 (40)
Now, let us try to set the coecients of the (C

)
2
terms to zero simultaneously. This
implies that
b
I
= 8p
I
(41)
for I = 1; 2; 3 and therefore that
p
1;2
= 0; p
3
= 33 (42)
From this one obtains two separate equations for the parameter  given by
 =
3
8
dimG+
1
8
+
1
24
X
1  90 (43)
for I = 1; 2 and
 =
3
8
dimG +
1
8
+
1
24
X
1 (44)
for I = 3. Clearly these two equations are incompatible and, hence, it is impossible to have
all vanishing (C

)
2
terms in the 1-loop corrected Lagrangian of Z
4
orbifolds. We nd that
the same results hold in other orbifolds as well.
7
4 Conclusion
We conclude that non-vanishing (C

)
2
terms are generated by light eld loops in the
N = 1, D = 4 Lagrangian of Z
N
orbifold superstrings. It is conceivable that loops containing
the heavy tower of states might cancel these terms, but we nd no reason, be it duality or
any other symmetry, for this to be the case [11]. This is presently being checked by a
complete genus-one string calculation [12]. We conjecture that cancellation will not occur.
Unfortunately, since 
T
, #
T
and ' are unknown, we can say nothing concrete about the
existence of R
2
terms in the Lagrangian. This issue will also be nally resolved in the
complete superstring calculation.
References
[1] K. Stelle, Gen. Rel. and Grav. 9 (1978) 353.
[2] B. Whitt, Phys. Lett. 145B (1984) 176.
[3] D. G. Boulware, G. T. Horowitz, and A. Strominger, Phys. Rev. Lett. 50 (1983) 1726;
E. Tomboulis, Phys. Lett. 97B (1980) 77.
[4] S. W. Hawking, in Quantum Field Theory and Quantum Statistics, eds. I. A. Batalin,
C. S. Isham, and G. A. Vilkovisky, A. Hilger (1987) 129.
[5] A. A. Starobinsky, Phys. Lett. 91B (1980) 99.
[6] A. Hindawi, B. A. Ovrut, and D. Waldram, in preparation.
[7] A. Hindawi, B. A. Ovrut, and D. Waldram, in preparation.
[8] G. L. Cardoso and B. A. Ovrut, Nuc. Phys. B 369 (1992) 351.
[9] G. L. Cardoso and B. A. Ovrut, Nuc. Phys. B 418 (1994) 535.
[10] J. P. Derindinger, S. Ferrara, C. Kounnas, and F. Zwirner, Nuc. Phys. B 372 (1992)
145.
[11] G. L. Cardoso, D. Lust, and B. A. Ovrut, Nuc. Phys. B 436 (1995) 65.
[12] K. Foerger, B. A. Ovrut, and S. Theisen, in preparation.
8


Higher-derivative gravitation in bosonic and superstring theories

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Higher-Derivative Gravitation in Bosonic and Superstring Theories

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